E2 (H4-lite) — dynamics near filaments vs the lifted-geometry premise

Setup: 4 PM N-body runs (128³ grid & particles, EdS, a: 0.1→1.0, 90 steps), 50k tracked particles per run over a ∈ [0.5, 1.0]. Spines: SE(3) lift on the final density; e3 = tidal minor eigenvector (filament axis). P1: mean

v̂·e3

at a=1 (null 0.5). P2: squared projection of the mid-path deviation from the straight chord onto e3 (null 1/3). dev = mean deviation magnitude (voxels = h⁻¹Mpc).

d to spine (vox)	n/seed	P1 ⟨|v̂·e3|⟩	P2 ⟨(d̂ev·e3)²⟩	⟨|dev|⟩ vox
0–2	20262	0.559 ± 0.013	0.364 ± 0.010	1.72
2–4	7244	0.537 ± 0.008	0.349 ± 0.004	1.22
4–8	6730	0.544 ± 0.011	0.327 ± 0.005	0.39
8–16	10315	0.572 ± 0.009	0.233 ± 0.005	0.27
16–64	5447	0.564 ± 0.020	0.208 ± 0.014	0.25

Nulls: P1 = 0.5, P2 = 1/3 ≈ 0.333 (isotropic). Values are across-seed mean ± sd; per-seed n is large so per-seed standard errors are negligible — seed scatter is the relevant uncertainty.